In many popular descriptions about the so-called "twin paradox" (which is not really a paradox) the message is that what makes the difference is that one of the twins is accelerated. And what many laymen learn from such descriptions is that it is that phase of the acceleration which makes the difference, which somehow leads to the difference in the age of the twins. Even Feynman has presented the twin paradox in such a way:

So the way to state the rule is to say that the man who has felt the accelerations, who has seen things fall against the walls, and so on, is the one who would be the younger; that is the difference between them in an "absolute" sense, and it is certainly correct.

Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics I, ch. 16 - 2 The twin paradox

This is not correct but completely misleading. And this can be easily shown, by the following variant of the twin paradox, where both twins have completely similar acceleration and deceleration events, but end up anyway differently aged.

Look at the picture at the right side: Here, both twins travel, at least a little bit. Only the red twin travels much longer time, and stays at the final place much shorter time. So, there is a long time when the red twin is travelling but the green twin is at rest. And this is what matters: The clock of the travelling twin is going slower. All imaginable clocks - inclusive his own aging as a biological clock. And therefore he ends up younger when they meet again.

This is simply what the Lorentz ether interpretation tells us. And in the Lorentz ether interpretation where is no such animal as a "twin paradox".

And this is also what the formula for "proper time" (the German original is "Eigenzeit", which translates more accurately as own, private, separate time) tells us: Acceleration does not matter at all, what matters is velocity: \[ \tau = \int \sqrt{1 - \frac{v^2(t)}{c^2}} dt\]

Let's note here also another nonsensical variant of using acceleration as a key to understand the twin "paradox": SR is about inertial systems, and does not handle acceleration. So, the twin paradox simply cannot be considered in SR, one needs accelerated systems of coordinates, thus, one needs GR. This is simply complete nonsense. One needs GR only when gravity is involved. SR can handle accelerated motion without any problem. The formula for proper time given above works for accelerated motion, where \(v(t)\neq const\), as well as for constant velocity.

But does that mean there is something wrong with the relativistic picture? For the travelling twin, tells the "twin paradox", it looks like the other twin moves, thus, his clock should be slower. This happens if he flies away, but also as he flies home. Thus, if we take everything together, it is the twin at home which will stay younger. Not?

Not. First of all, it is not what it "looks like". The very idea that all this is what the moving observers "sees" is wrong - he makes a theoretical decision to use Einstein synchronization, which is, essentially, a hypothesis, namely that he is actually at rest.

The point which is missed in this description is what really changes if the travelling twin makes his turn. This looks like changing nothing - he remains on the same place. But the misleading "looks like" hides that he makes a switch from one inertial frame to another inertial frame. Which is something very unnatural. Either he thought that he was at rest before the turn, or that he is at rest after the turn - but he cannot be at rest before and after the turn. Ok, he may have changed his opinion. But that change of opinion also changes a lot of other things, in particular also what is the time of "now" on Earth for the turning point event.

Once this "now" changes, this change has to be taken into account. The difference between these two different opinions what is "now" makes the difference, and if it is forgotten, the result will be wrong. It is easy to forget it because in our everyday experiences we do not change our opinion what is "now". We live in a world where "now" makes sense as defined by Nature, independent of our behaviour. And, so, we would not change, if we would travel, our opinion what is "now" for the twin who remains at home. There would be no point doing this.

The problem behind this is the use of Einstein synchronization to define what is now.

The easy way to see this is if one assumes the Lorentz ether. Then, there is an absolute, correct notion of "now", defined by absolute time being equal. And this correct "now" is also the one established by Einstein synchronization if an observer is at absolute rest. For every moving observer, Einstein synchronization gives a wrong, distorted result.

But, given relativistic symmetry, which holds in the Lorentz ether for everything observable, there is no way to identify absolute rest by observation. What can we do? We can simply make a (possibly wrong) hypothesis about what is absolute rest. And then, all we have to do is to follow this single arbitrary choice consistently. As long as we remain consistent in our possible error, everything is fine. We assume that, say, we are now in rest, and assume this for everything - time dilation, length contraction, Einstein synchronization - the error will not become visible. This follows from the simple point that if it would somehow become visible, we would be able to observe that we are now not in rest, and would be, as a consequence, able to identify absolute rest.

If we would follow this rule in the twin paradox, it appears that there is no problem. Say, the twin at home is really at rest. Once we have no way to know this, we can assume that the twin travelling away from home is at rest. That would be wrong, but so what. From this point of view, it looks like he is older than the twin at home at the turning point, But after this the twin travelling home has two times the speed, and relativistic time dilation becomes even stronger, so that the final result is the same as in the correct picture - the twin staying at home is older. A similar picture, with the same result, follows if we assume that the twin travelling home is at rest.

But what would be obviously stupid would be to assume during the first part of the trip that the twin travelling away from home is at rest, and during the second part that the twin travelling home is at rest. There is only one rest, even if we don't know which it is, it does not change in time. Our hypothesis about what is rest may change - but if this happens, we automatically have to reevaluate all the conclusions we have made before based on the old, now rejected hypothesis.

The situation looks less clear in the spacetime interpretation. Here, we have no correct notion of absolute rest, all notions of contemporaneity are equally valid. Instead of all except possibly one frame having wrong contemporaneity, they are, somehow, all correct, equally correct, just somehow different. What would, in this case, prevent us from switching from one correct description to another, equally correct one?

What makes the difference is that in the Lorentzian picture both frames may be wrong, may be true, but at least one of them is certainly wrong, thus, switching from one to another one is clearly wrong. In the spacetime view, they are somehow all "correct" in the same way. How to see that using one correct frame during the first part and a different correct frame during the second part gives a nonsensical answer?

Of course, for mathematicians this is not such a serious problem. They know how to handle different systems of coordinates. The frames are, from the position of the mathematician, not wrong. All what is problematic is that the hyperplane \(t=0\) is not the space, but some strange skew plane in spacetime. For the mathematician, this is not a problem at all. He knows how to handle such skew frames, and knows how to switch from one skew frame to another skew frame. One has to look at the formulas, that's all.

The difference is, thus, not about mathematics - the mathematics is the same in both interpretations. The difference is in the intuitions. The spacetime interpretation removes the intuition that one has to consider the whole story in a single frame to remain consistent. And to switch from one frame to another one requires to care about the changing notion of "now", which nothing one would have to care in everyday life.